Vladimir A. Volpert

Traveling Wave Solutions of Parabolic Systems

Part I. Stationary waves: Scalar equation Leray-Schauder degree Existence of waves Structure of the spectrum Stability and approach to a wave Part II. Bifurcation of waves: Bifurcation of nonstationary modes of wave propagation Mathematical proofs Part III. Waves in chemical kinetics and combustion: Waves in chemical kinetics Combustion waves with complex kinetics Estimates and asymptotics of the speed of combustion waves Asymptotic and approximate analytical methods in combustion problems (supplement) Bibliography.

Forced forward smolder combustion

Abstract We consider porous cylindrical samples closed to the surrounding environment except at the ends, with gas forced into the sample through one of the ends. A smolder wave is initiated at that end and propagates in the same direction as the flow of the gas. We employ asymptotic methods to find smolder wave solutions with two different structures. Each structure has two interior layers, i.e., regions of relatively rapid variation in temperature separated by longer regions in which the temperature is essentially constant. One layer is that of the combustion reaction, while the other is due to heat transfer between the solid and the gas. The layers propagate with constant, though not necessarily the same, velocity, and are separated by a region of constant high temperature. A so-called reaction leading wave structure occurs when the velocity of the combustion layer exceeds that of the heat transfer layer, while a so-called reaction trailing wave structure is obtained when the combustion layer is slower than the heat transfer layer. The former (latter) occurs when the incoming oxygen concentration is sufficiently high (low). Reaction trailing structures allow for the possibility of quenching if the gas mass influx is large enough; that is, incomplete conversion can occur due to cooling of the reaction by the incoming gas. For each wave structure there exist stoichiometric, and kinetically controlled solutions in which the smolder velocity is determined, respectively, by the rate of oxygen supply to the reaction site and by the rate of consumption in the reaction, i.e., by the kinetic rate. Stoichiometric (kinetically controlled) solutions occur when the incoming gas flux is sufficiently low (high). For each of the four solution types, we determine analytical expressions for the propagation velocities of the two layers, the burning temperature, and the final degree of solid conversion. We also determine analytical expressions for the spatial profiles of temperature, gas flux, and oxygen concentration. Gravitational forces are considered and are shown to have a minimal effect provided the ambient pressure is large compared to the hydrostatic pressure drop. The solutions obtained provide qualitative theoretical descriptions of various experimental observations of forward smolder. In particular, the reaction trailing stoichiometric solution corresponds to the experimental observations of Ohlemiller and Lucca, while the reaction leading stoichiometric solution corresponds to the experimental observations of Torero et al.

Propagation and extinction of forced opposed flow smolder waves

Abstract Smoldering is a slow combustion process in a porous medium in which heat is released by oxidation of the solid. If the material is sufficiently porous to allow the oxidizer to easily filter through the pores, a smolder wave can propagate through the interior of the solid. We consider samples closed to the surrounding environment except at the ends, with gas forced into the sample through one of the ends. A smolder wave is initiated at the other end and propagates in a direction opposite to the flow of the oxidizer. Previous experimental results show that for opposed flow smolder, decomposition of the solid fuel into char is the chemical process which drives the smolder process. We model this decomposition as a one step reaction. The model suggests that extinction occurs when decomposition is complete. We employ large activation energy asymptotic methods to find uniformly propagating, planar smolder wave solutions. We determine their propagation velocity, burning temperature, final degree of fuel decomposition, and extinction limits. We also determine spatial profiles of gas flux, oxidizer concentration, temperature, and degree of decomposition of the solid. Comparison is made with previous experimental results.

Turing pattern formation in the brusselator model with superdiffusion

The effect of superdiffusion on pattern formation and pattern selection in the Brusselator model is studied. Our linear stability analysis shows, in particular, that, unlike the case of normal diffusion, the Turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator. A weakly nonlinear analysis yields a system of amplitude equations, analysis of which predicts parameter regimes where hexagons, stripes, and their coexistence are expected. Numerical computations of the original Brusselator model near the stability boundaries confirm the results of the analysis. In addition, further from the stability boundaries, we find a regime of self-replicating spots.

Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion

Abstract We consider the evolution of a gasless combustion front. We derive coupled complex Ginzburg-Landau type equations for the amplitudes of waves along the front as functions of slow temporal and spatial variables. The equations are written in characteristic variables and involve averaged terms which reflect the fact that in the slowest time scale, the effect on one wave, of a second wave traveling with the group velocity in the opposite direction on the intermediate time scale, enters only through its average. Solutions of the amplitude equations in the form of traveling, standing, and quasiperiodic waves are found, and regions of stability for these solutions are determined. In particular we find that the traveling and quasiperiodic (including standing) waves are not stable simultaneously. Finally, we observe that the stability analysis for coupled complex Ginzburg-Landau equations with averaged terms differs from that for coupled complex Ginzburg-Landau equations with the averages removed.

On the steady-state approximation in thermal free radical frontal polymerization

Frontal polymerization is a process in which a spatially localized reaction zone propagates into a monomer converting it into a polymer. We formulate a mathematical model of thermal free radical frontal polymerization involving a five-species reaction mechanism and justify its reduction to a four-species problem by combining the concentrations of two kinds of radicals, primary radicals and polymer radicals. We study the simplified problem both numerically and analytically, and determine such important characteristics of the process as the propagation speed of the wave, the final temperature in the system, and the spatial distributions of all the species in the wave. We perform the analyses with the steady-state approximation (SSA) and without it, and reveal where the SSA solution breaks down. Specifically, the non-SSA solution predicts full monomer conversion and an adiabatic final system temperature, whereas the SSA solution predicts a less-than-adiabatic final temperature and incomplete conversion.

Self-Compaction or Expansion In Combustion Synthesis of Porous Materials

Abstract-We propose a mathematical model for the combustion of porous deformable condensed materials, which we use to describe the deformation of the high temperature products, induced by the pressure difference of the gas outside and inside the sample, in the absence of any external forces. The deformation occurs as a result of pore compaction (expansion), resulting in a more (less) dense product material. To describe the evolution of porosity we derive an equation which allows us to define a characteristic time ofdeformation fJ. If td is sufficiently smaller than the characteristic time of combustion t, the deformation process is sufficiently fast to compensate for pressure gradients. so that pressure is equalized almost instantaneously, and filtration is suppressed. If td > tr deformation occurs solely in the product, and does not affect the propagation velocity. We determine various characteristics of a uniformly propagating combustion wave, and the materials produced by it, such as the propagation ve...

Fronts in anomalous diffusion–reaction systems

A review of recent developments in the field of front dynamics in anomalous diffusion–reaction systems is presented. Both fronts between stable phases and those propagating into an unstable phase are considered. A number of models of anomalous diffusion with reaction are discussed, including models with Lévy flights, truncated Lévy flights, subdiffusion-limited reactions and models with intertwined subdiffusion and reaction operators.

Nonadiabatic frontal polymerization

Frontal polymerization is a process in which a spatially localized reaction zone propagates into a monomer, converting it into a polymer. In the simplest case of free-radical polymerization, a mixture of a monomer and initiator is placed into a test tube. Upon reaction initiation at one end of the tube a self-sustained thermal wave, in which chemical conversion occurs, develops and propagates through the tube. In a previous paper, a perfectly insulated tube (i.e., an adiabatic polymerization process) was considered. In reality, it is nearly impossible to eliminate heat losses completely, and an accurate model must take this into account. Extinction of polymerization waves and difficulties initiating the wave, both as a result of heat losses, are often encountered in experiments. This paper will therefore concentrate on nonadiabatic frontal polymerization. The propagation of nonadiabatic free-radical polymerization fronts is studied by methods originally developed in combustion theory, and employed in a previous paper. This analysis is accomplished by examination of the structure of the polymerization wave, its propagation velocity, degree of conversion of the monomer and maximum temperature, and how these quantities are affected by changes in initial temperature, concentrations and kinetic parameters. The values of these quantities near the extinction limit (beyond which traveling-wave solutions will no longer exist) are compared to those in the adiabatic case.

Nonlinear dynamics in a simple model of solid flame microstructure

Abstract We present a model of condensed phase combustion which attempts to elucidate the effects of spatially localized reaction sites on the propagation of a combustion wave. Heat transfer is assumed to be uniform but exothermic reactions are allowed to occur only at evenly distributed locations, that we refer to as reactant particles. Thus, combustion wave propagation manifests itself as a process of sequential ignition and burning of particles. Green’s functions are used to show that “steady” wave speed is related to particle ignition temperature, particle geometry and the ratio of heat diffusion to reaction times through a single transcendental equation. Furthermore, for the one-dimensional case, the dynamics of this system can be related to a history dependent implicit map f → : R ∞ → R ∞ which determines time to the next ignition. Iteration of this map demonstrates that average wave speed undergoes a period doubling bifurcation to chaos and subsequent extinction. A linear stability analysis of this map is performed to determine the stability boundaries for period 2n orbits. Additionally, temperature profiles are shown to be in qualitative agreement with experiments which describe a transition from the so-called quasi-homogeneous to relay-race regimes.